A next generation Digital Subscriber Line (DSL) system, G.fast, is currently being standardized by the International Telecommunication Union-Telecommunication Standardization Sector (ITU-T). Current specifications have been drafted as ITU-T G.9700 and G.9701. G.fast is also referred to as a Fiber-to-the-Distribution-Point (FTTdp) system, where Digital Subscriber Line Access Multiplexers (DSLAMs) are located in the last distribution point in the copper loop, up to 250 meters away from the end users. At such short distances, much higher frequencies should be used, e.g. up to 200 Megahertz (MHz) or more. For these higher frequencies, especially higher than 100 MHz, when multiple lines are located together, the crosstalk levels relative to the direct channel gain are considerably increased compared to Very-high-bit-rate DSL 2 (VDSL2) frequencies (up to 30 MHz), and therefore an efficient crosstalk cancellation scheme is vital.
One way to reduce crosstalk among lines is vectoring. Vectoring was first standardized in 2010 for VDSL2 in the ITU-T G.993.5 recommendation to cancel out crosstalk below 30 MHz. In VDSL2, linear cancellation schemes based on the Zero-Forcing (ZF) criterion are used on the DSLAM side with a linear precoder for downstream and a linear crosstalk canceler in upstream. It has been shown that the linear cancellation schemes specified in ITU-T G.993.5 approach the crosstalk-free single-line performance (i.e., the performance a line would achieve if it were not located with other lines). The reason the linear schemes achieve the near-optimal performance is that a channel matrix, H, is Row-Wise Diagonal Dominant (RWDD) and column-wise diagonal dominant (CWDD) in downstream and upstream, respectively, hence the crosstalk channel is much lower than the direct channel.
In the G.fast standardization, it has been agreed to have vectoring mandatory in order to transmit at high frequencies and achieve a very high data rate of up to 1 Gigabit Per Second (Gbps) aggregated. However, the linear cancellation schemes are not good to use in G.fast, especially for downstream transmission. The channel matrix, H, at high frequencies in G.fast presents much higher crosstalk relative to the direct channel gain than at VDSL frequencies. In measurements, the crosstalk channel has been shown, in some occasions, to be even stronger than the direct channel in high frequencies. This strong crosstalk results in the output signal per port being amplified considerably by the linear precoder. Power normalization at each tone is required to keep the signal power below the Power Spectral Density (PSD) mask. The power normalization degrades the performance significantly when crosstalk is high in higher frequencies.
Therefore, it has been agreed in G.fast that in phase 1 (up to 106 MHz frequency), a linear precoder should be used. In phase 2 (upgrading to support up to 212 MHz), the non-linear Tomlinson-Harashima Precoder (THP) should be used. Unlike the ZF-based linear precoder, which inverts the channel matrix, H, in order to cancel crosstalk, the non-linear precoder is based on the QR decomposition of the conjugate transpose of the channel matrix:H*=QR  (1)where Q is an N×N unitary matrix and R is an upper triangular matrix. The transmitted signal vector x is defined as:x=Qx′  (2)where x′ is an N×1 vector, representing a precoded version of the original signal vector {tilde over (x)}. Denoting the noise by n, the received signal y is:y=Hx+n  (3)y=HQx′+n  (4)y=R*Q*Qx′+n  (5)y=R*x′+n  (6)
The received signal will be crosstalk free when:S{tilde over (x)}=R*x′  (7)where S is an N×N diagonal matrix in which the diagonal elements are equal to the diagonal elements of R*. Solving equation (7), the precoding operation is defined as:
                              x          1          ′                =                              x            ~                    1                                    (        8        )                                                      x            2            ′                    =                                                    x                ~                            2                        -                                                            r                                      2                    ,                    1                                                                    r                                      2                    ,                    2                                                              ⁢                              x                1                ′                                                    ⁢                                  ⁢        ⋮                            (        9        )                                          x          N          ′                =                                            x              ~                        N                    -                                                    r                                  N                  ,                                      N                    -                    1                                                                              r                                  N                  ,                  N                                                      ⁢                          x                              N                -                1                            ′                                -          ⋯          -                                                    r                                  N                  ,                  1                                                            r                                  N                  ,                  N                                                      ⁢                          x              1              ′                                                          (        10        )            where x′i denotes the elements of x′, {tilde over (x)}i represent the elements of {tilde over (x)} and ri, j are the elements of R.
The precoding operation in equations (8) through (10) may lead to a significant increase in the energy of x′ compared to x. To avoid this, a modulo arithmetic operation is applied to the elements in equations (8) through (10). The modulo arithmetic operation is defined as follows:
                                          Γ                          M              i                                ⁡                      [            x            ]                          =                  x          -                                    M              i                        ⁢                          ⌊                                                x                  +                                                            M                      i                                        /                    2                                                                                        M                    i                                    ⁢                  d                                            ⌋                                                          (        11        )            where x is the received symbol from a Pulse-Amplitude Modulation (PAM) constellation, d is the distance between symbols in the PAM constellation and Mi is the PAM constellation order. Viewing Quadrature-Amplitude Modulation (QAM) as two orthogonal PAM constellations, the corresponding modulo operation is defined as Γ√{square root over (Mi)}[x]=Γ√{square root over (Mi)}[(x)]+jΓ√{square root over (Mi)}[(x)].
                              x          1          ′                =                              x            ~                    1                                    (        12        )                                                      x            2            ′                    =                                    Γ                              M                2                                      ⁡                          [                                                                    x                    ~                                    2                                -                                                                            r                                              2                        ,                        1                                                                                    r                                              2                        ,                        2                                                                              ⁢                                      x                    1                    ′                                                              ]                                      ⁢                                  ⁢        ⋮                            (        13        )                                          x          N          ′                =                              Γ                          M              N                                ⁡                      [                                                            x                  ~                                N                            -                                                                    r                                          N                      ,                                              N                        -                        1                                                                                                  r                                          N                      ,                      N                                                                      ⁢                                  x                                      N                    -                    1                                    ′                                            -              ⋯              -                                                                    r                                          N                      ,                      1                                                                            r                                          N                      ,                      N                                                                      ⁢                                  x                  1                  ′                                                      ]                                              (        14        )            
Mathematically, the received signal can be expressed as yi=ri,i{tilde over (x)}i,i+ni. At the receiver, the same modulo operation is applied to recover the signal:
                                          y            ^                    i                =                              Γ                          M              i                                ⁡                      [                                          y                i                                            r                                  i                  ,                  i                                                      ]                                              (        15        )            Therefore, the Signal-to-Noise Ratio (SNR) of the i-th received signal is obtained as:
                              SNR          THP                =                                                            ξ                ~                            i                        ⁢                                                                            r                                      i                    ,                    i                                                                              2                                            σ            i            2                                              (        16        )            where {tilde over (ξ)}i is the average energy of {tilde over (x)}i and σi2 the noise variance.
After the modulo arithmetic operation, the transmitted signal may still suffer an energy increase, which for square-QAM constellations is Mi/(Mi−1), where Mi is the QAM modulation order for line i. For non-square QAM constellations, the relationship is more irregular. In G.fast: Constellations for use with Non-Linear Pre-coding, Ikanos, Switzerland: Contribution ITU-T 2013-01-Q4-02, 2013, the calculation of the energy factors for all the QAM constellations used in VDSL2 in the range 4 to 4096 is presented. The energy increasing factors are summarized in Table 1. Table 1 shows that the even-bit QAM modulation orders have much lower energy increases than those of the odd-bit QAM modulation orders.
TABLE 1Energy increase in dB for the THPModulation order (b bits)Energy increase in dB4(2)1.258(3)2.5016(4)0.28032(5)0.7964(6)0.0684128(7)0.68256(8)0.0170512(8)0.661024(9)0.00422048(10)0.654096(11)0.0011
Although THP non-linear precoding achieves higher system capacity than the ZF-based linear precoder, there is still a need for systems and methods for determining non-linear precoding coefficients.